Optimal. Leaf size=49 \[ \frac {1}{3} \sqrt {x-1} \sqrt {3 x+5}-\frac {8 \sinh ^{-1}\left (\frac {1}{2} \sqrt {\frac {3}{2}} \sqrt {x-1}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1958, 50, 54, 215} \[ \frac {1}{3} \sqrt {x-1} \sqrt {3 x+5}-\frac {8 \sinh ^{-1}\left (\frac {1}{2} \sqrt {\frac {3}{2}} \sqrt {x-1}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rule 1958
Rubi steps
\begin {align*} \int \sqrt {\frac {-1+x}{5+3 x}} \, dx &=\int \frac {\sqrt {-1+x}}{\sqrt {5+3 x}} \, dx\\ &=\frac {1}{3} \sqrt {-1+x} \sqrt {5+3 x}-\frac {4}{3} \int \frac {1}{\sqrt {-1+x} \sqrt {5+3 x}} \, dx\\ &=\frac {1}{3} \sqrt {-1+x} \sqrt {5+3 x}-\frac {8}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {8+3 x^2}} \, dx,x,\sqrt {-1+x}\right )\\ &=\frac {1}{3} \sqrt {-1+x} \sqrt {5+3 x}-\frac {8 \sinh ^{-1}\left (\frac {1}{2} \sqrt {\frac {3}{2}} \sqrt {-1+x}\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 76, normalized size = 1.55 \[ \frac {3 (x-1) \sqrt {3 x+5}-8 \sqrt {3} \sqrt {x-1} \sinh ^{-1}\left (\frac {1}{2} \sqrt {\frac {3}{2}} \sqrt {x-1}\right )}{9 \sqrt {\frac {x-1}{3 x+5}} \sqrt {3 x+5}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 54, normalized size = 1.10 \[ \frac {1}{3} \, {\left (3 \, x + 5\right )} \sqrt {\frac {x - 1}{3 \, x + 5}} + \frac {4}{9} \, \sqrt {3} \log \left (\sqrt {3} {\left (3 \, x + 5\right )} \sqrt {\frac {x - 1}{3 \, x + 5}} - 3 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 74, normalized size = 1.51 \[ -\frac {8}{9} \, \sqrt {3} \log \relax (2) \mathrm {sgn}\left (3 \, x + 5\right ) + \frac {4}{9} \, \sqrt {3} \log \left ({\left | -\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2 \, x - 5}\right )} - 1 \right |}\right ) \mathrm {sgn}\left (3 \, x + 5\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 2 \, x - 5} \mathrm {sgn}\left (3 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 76, normalized size = 1.55 \[ -\frac {\sqrt {\frac {x -1}{3 x +5}}\, \left (3 x +5\right ) \left (4 \sqrt {3}\, \ln \left (\sqrt {3}\, x +\frac {\sqrt {3}}{3}+\sqrt {3 x^{2}+2 x -5}\right )-3 \sqrt {3 x^{2}+2 x -5}\right )}{9 \sqrt {\left (3 x +5\right ) \left (x -1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.39, size = 80, normalized size = 1.63 \[ \frac {4}{9} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 3 \, \sqrt {\frac {x - 1}{3 \, x + 5}}}{\sqrt {3} + 3 \, \sqrt {\frac {x - 1}{3 \, x + 5}}}\right ) - \frac {8 \, \sqrt {\frac {x - 1}{3 \, x + 5}}}{3 \, {\left (\frac {3 \, {\left (x - 1\right )}}{3 \, x + 5} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.16, size = 57, normalized size = 1.16 \[ -\frac {8\,\sqrt {3}\,\mathrm {atanh}\left (\sqrt {3}\,\sqrt {\frac {x-1}{3\,x+5}}\right )}{9}-\frac {8\,\sqrt {\frac {x-1}{3\,x+5}}}{3\,\left (\frac {3\,x-3}{3\,x+5}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {x - 1}{3 x + 5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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